The natural volume method (NVM): Presentation and application to shallow water inviscid flows

Abstract

SUMMARYIn this paper a fully Lagrangian formulation is used to simulate 2D shallow water inviscid flows. The natural element method (NEM), which has been used successfully with several solid and fluid mechanics applications, is used to approximate the fluxes over Voronoi cells. This particle-based method has shown huge potential in terms of handling problems involving large deformations. Its main advantage lies in the interpolant character of its shape function and consequently the ease it allows with respect to the imposition of Dirichlet boundary conditions.In this paper, we use the NEM collocationally, and in a Lagrangian kinematic description, in order to simulate shallow water flows that are boundary moving problems. This formulation is ultimately shown to constitute a finite-volume methodology requiring a flux computation on Voronoi cells rather than the standard elements, in a triangular or quadrilateral mesh. St Venant equations are used as the mathematical model. These equations have discontinuous solutions that physically represent the existence of shock waves, meaning that stabilization issues have thus been considered. An artificial viscosity deduced from an analogy with Riemann solvers is introduced to upwind the scheme and therefore stabilize the method. Some inviscid bidimensional flows were used as preliminary benchmark tests, which produced decent results, leading to well-founded hopes for the future of this method in real applications.

A Weighted Average Flux (WAF) scheme applied to shallow water equations for real-life applications

“…(33) or (31) directly will be sufficient, theoretically, to guarantee a second order in time. However, in order to further improve time integration and for the sake of stability, we can use a Newmark [10,27] scheme given by:…”

“…2). Another way to define these cells is to use the Voronoi diagram [10], but in this case a Delaunay mesh must be used. Although this description leads to additional preprocessing, it has the advantage of being much less sensitive to mesh quality [28].…”

Meshless method for shallow water equations with free surface flow

“…Among the newest meshless methods, the Natural element method (NEM) presents some interesting features [1,15,16]. Also, it has been recently demonstrated [7] that the NEM possesses a particularly well-suited structure for the application of stabilized conforming nodal integration schemes [5], thus leading to a very accurate nodal method with great accuracy in numerical integration.…”

Extension of the natural element method to surface tension and wettability for the simulation of polymer flows at the micro and nano scales

“…A complete literature review of the NEM is beyond this paper, but the work of Cueto et al [7] about non convex boundaries in solid mechanics and its extensions to the flows of complex fluids by Martínez et al [8] and to fluid dynamics by González et al [9] are worth noting in connection with the present work. Shallow water flows have also been simulated with the NEM by Ata et al [10], who applied the technique developed by Yvonnet et al [11] for non convex boundaries, and by Darbani et al [12]. Moreover, surface tension has been included by Defauchy et al [13] recently, for application to polymer powder sintering, but without fluid-solid contact.…”